The composite-particle scenario is a phenomenology that can organize the data of the ''sharp lepton problem'' posed by heavy-ion and (beta(+) + atom) studies. It hypothesizes a new composite particle (of mass similar to 3mc(2)) as the source of the observed sharp energy (e(+)e(-)) decay pairs. Available data rule out the possibilities that the source is a new elementary particle or that it is a quasi-bound state of (e(+)e(-)). Occam's razor therefore currently favors the quadronium structure, Q(0) = (e(+)e(+)e(-)e(-)). Implications of quadronium for high-precision quantum electrodynamics (QED) are considered, and calculated and (or) measured deviations in QED that are sensitive to the existence of Q(0) are identified. In particular, for the electron magnetic-moment anomaly, a(e) = (g(c) - 2)/2, a Q(0)-pole effects a small correction to the contributions of O(alpha(4)), which is therefore small compared to the largest current (theoretical) uncertainty. For photon-photon scattering, Q(0) corrects the leading order matrix element, and allows resonant Q(0) creation in photon-nucleus scattering. Finally, a Q(0) bound state corrects the O(alpha) correction to the leading 3 gamma annihilation rate of triplet positronium. Therefore Q(0) may contribute significantly to this decay rate, which is currently in a 10 sigma discrepancy with experiment. A current experimental gap is the lack of corroborative data on the sharp (Gamma less than or equal to 2.1 keV) 330.1 keV electrons reported by Sakai from irradiations of U and Th with beta(+)-decay positrons. A study of these (and (or) their expected partner positrons of the same energy) in collisions of (similar to 3 MeV) beam positrons (or electrons) upon high-Z neutral atoms could fill this gap. Similar studies with positrons of 660-795 keV would test the expectation that recoilless resonance creation of the Q(0) source of these pairs is also possible.
